5. In addition to mass and distance, let us now consider time as another
fundamental unity. To do so, let us try to define time in terms of the
same unit, the proton. Speed or momentum is distance divided by time, so
if we can find a fundamental "speed" which we can equate with our
fundamental distance (the Compton wavelength of the proton), then we
will have all terms in our equation, and we will be able to develop a
fundamental unit of time from these.

The "natural" unit of speed is the speed of light (c), a fundamental
constant of nature. If distance/time = speed, then a fundamental unit of
distance divided by a fundamental unit of time may equal the speed of
light. If the premises developed above have any significance, we should be
able to divide the wavelength of the proton (lambda_p) by a fundamental
unit of time (t) to derive the speed of light (c). In symbolic terms,

lambda_p / t = c

or

t = lambda_p / c

Again using the CODATA values, we have

t = lambda_p / c = 1.32141 x 10^-13 cm / 2.9979246 x 10^10 cm/sec

and, therefore,

t = 4.407749 x 10^-24 sec

We will call this unit of time the "time of a proton," or time_p, or simply
t_p. This result (t_p = 4.407749 x 10^-24 sec) approximates the
experimentally determined time scale for nuclear interactions -- often
stated roughly as 10^-24 sec.

6. Using this unit of time (t_p), we can state that one second equals a
certain number of t's, i.e.,

sec = t_p / 4.407749 x 10^-24 = 2.2687317 x 10^23 t_p

7. Using these units of mass, distance and time, let us imagine a number
line where negative numbers represent all distances and positive numbers
represent all masses, each expressed in units of time. To reiterate,

8. In his book, A Brief History of Time, Stephen Hawking invokes the
concept of "imaginary time" for the purposes of certain quantum
mechanical calculations (at p. 134), and relativistic effects (p. 139). One
knot which this concept cuts through is the idea of time running forward
or backward (p. 143), which makes "imaginary time" particularly useful in
physics calculations which themselves make no distinction between
forward and backward time, despite our experience of the "arrow of time"
which travels exclusively forward.

"Imaginary time," in this sense, is not "fanciful time," but merely time
calculations which make use of "imaginary numbers" which incorporate
the square root of minus one (sqrt(-1)) -- a useful mathematical
device with an ancient pedigree.

Let us suppose that we may represent time itself as a "negative"
quantity, related to the square root of minus one. In order to relate this
quantity to our experience, we may deal with the absolute value of this
quantity, so that "time" is the absolute value of the square root of minus
one, or |sqrt(-1)|.

t = 4.407749 x 10^-24 sec = |sqrt(-1)|.

Rather than the absolute value, we also may set "t" as a positive number
by representing it as follows:

t = -(sqrt(-1))

Since -sqrt(-1) = 1 / +(sqrt(-1), we may further state that

t = 1 / -t
t = -1 / t

and-

t = 1 / t

Finally, we see that this formula further yields the result

t^2 = -1

II. Neutrinos

9. Neutrinos, as we understand them, are integral to the weak force or
interaction. Such interactions typically are known to take approximately
10^-10 seconds. Using just the units of mass, distance and time derived
above, we may further derive a "time of the neutrino" (tn), as we did the
"time of the proton" (t_p).

which is close to the "roughly 10^-10 sec" mentioned above for nuclear
interactions (paragraph 5).

10. Because the neutrino is closely associated with these weak nuclear
interactions, we will infer that this time interval is associated with the
neutrino itself. Therefore, we will denominate this time interval as the
"neutrino time," or t_n.

11. We saw a relationship between time and the mass of the proton, in
that the "proton time" divided by itself equaled the "proton mass."
(Mentioned above.) Thus, we may state

delta-t_p / delta-t_p = +1 = mass_p

Let us hypothesize that any time interval, divided by any other time
interval, will similarly represent some mass, as a multiple or fraction of
the mass of a single proton. (Here, we exclude "negative time"
speculations.)

delta-t_x / delta-t_y = k(+1) = a mass

where k is a positive number, whole or fractional.

Thus, time divided by time equals a mass -- a mass multiple -- a multiple
of the proton's mass, which is one.

12. Therefore, if the neutrino's time (t_n) is as set forth in paragraphs 9 and
10, above, we should be able to divide the neutrino time by itself to get a
mass associated with the neutrino (mass_n). This would work as follows:

mass_n = (gm.cm. / sec) / sec

= gm.cm. / sec^2

which may also be expressed

(cm / sec^2) gm

13. With the values derived above for gm, cm, and sec, we can derive a
mass associated with the neutrino as follows:

Thus, this method for correlating time to mass yields a neutrino mass of
1.470266 x 10^-34 gm.

This is a very small value for the mass of a particle, being many orders of
magnitude less than the mass of an electron, which is to be expected given
the difficulty of determining whether the neutrino had any mass at all.

14. As reported in the New York Times (June 5, 1998), recent experiments
at the Super-Kamiokande detector, while not yielding a value of any
neutrino masses, do "suggest that the difference between the masses of
muon neutrinos and other types of neutrinos [in the oscillations] is only
about 0.07 electron volts (a measure of particle mass)."

Mass may be expressed in electron volts (eV) because mass and energy are
equated through e = mc^2. According to the CODATA values (Table 4),

1 eV = 1.78266270 x 10^-36 kg, or 1.78266270 x 10^-33 gm.

Therefore, 0.07 eV = 1.24786389 x 10^-34 gm.

By equating a particle's mass with the time of its interactions, we come
up with a number that is comparable to the experimentally determined
oscillation mass of the neutrinos. Because of the uncertainty in the
reported experiments, it appears that our calculated value is at least in
the right ball park. As experiments are refined, I expect that the
experimentally determined value will approach more closely the
calculated value.